Convergence results for a coarsening model using global linearization
Thierry Gallay
Institut Fourier Universit´e de Grenoble I
BP 74
F-38402 Saint-Martin d’H`eres
Alexander Mielke
Institut f¨ur Analysis, Dynamik und Modellierung
Universit¨at Stuttgart Pfaffenwaldring 57 D-70569 Stuttgart
December 12, 2002
Abstract
We study a coarsening model describing the dynamics of interfaces in the one- dimensional Allen-Cahn equation. Given a partition of the real line into intervals of length greater than one, the model consists in repeatedly eliminating the shortest interval of the partition by merging it with its two neighbors. We show that the mean-field equation for the time-dependent distribution of interval lengths can be explicitly solved using a global linearization transformation. This allows us to derive rigorous results on the long-time asymptotics of the solutions. If the average length of the intervals is finite, we prove that all distributions approach a uniquely deter- mined self-similar solution. We also obtain global stability results for the family of self-similar profiles which correspond to distributions with infinite expectation.
1 Introduction
Consider a domain D ⊂ Rn which is divided into a large number of subdomains (or cells) of different sizes, separated by domain walls, and assume that the system evolves in such a way that the larger subdomains grow with time while the smaller ones shrink and eventually disappear. In particular, the average size of the cells increases, so that the subdivision of D becomes rougher and rougher. Such a coarsening dynamics is observed in many physical situations, especially near a phase transition when a system is quenched from a homogeneous state into a state of coexisting phases. Typical examples are the formation of microstructure in alloy solidification [LiS61, KoO02] and the phase separation in lattice spin systems [De97, KBN97]. Closely related to coarsening is the coagulation (or aggregation) process which describes the dynamics of growing and coalescing droplets [DGY91, PeR92, Vo85]. In this case, the system consists of a large number of particles of different masses which interact by forming clusters. Again, the total mass is preserved, so that the average mass per cluster increases with time.
Given a coarsening or a coagulation model, the main task is to predict the long-time evolution of the size distribution of the cells, or the mass distribution of the clusters. In many cases, experiments and numerical calculations show that this behavior is asymp- totically self-similar: the system can be described by a single length scale L(t), and the distribution approaches the scaling form L(t)−1Φ(x/L(t)) as t → ∞. The profile Φ and the asymptotics of L(t) can sometimes be determined exactly [NaK86, BDG94]. How- ever, even in simple situations, it is very difficult to prove that the distribution actually converges to a self-similar profile.
In this work, we consider a simple coarsening model related to the one-dimensional Allen-Cahn equation ∂tu =∂x2u+ 12(u−u3), where x ∈R. The equilibria of this system are the homogeneous steady states u=±1, together with the kinks u(x) =±tanh(x/2) which represent domain walls separating regions of different “phases”. Ifuis any bounded solution of this equation, then fort >0 sufficiently large the graph of u(t,·) will typically look like a (countable) family of kinks separated by large intervals on which u ≈ ±1. If we denote by xj(t) the position of thejth kink and if we assume that xj+1(t)−xj(t)1 for allj ∈Z, a rigorous asymptotic analysis shows that ˙xj ≈F(xj+1−xj)−F(xj−xj−1), where F(y) = 24e−y [CaP89]. In other words, the positions of the domain walls behave like a system of point particles with short range attractive pair interactions. Thus, on an appropriate time scale, only the closest pairs of kinks will really move; in such pairs, kinks will attract each other until they eventually annihilate.
This kink dynamics suggests the following coarsening model [NaK86, DGY91, CaP92, BDG94, RuB94, BrD95, CaP00]. Consider a partition of the real line R into a countable union of disjoint intervals Ij, with `(Ij) ≥ 1 for all j ∈ Z. In the previous picture, the intervalsIj correspond to regions whereuis close to±1. A dynamics on this configuration space is defined by iterating the following coarsening step: choose the “smallest” interval in the partition, and merge it with its two nearest neighbors. This model clearly mimics the dynamics of the domain walls in the one-dimensional Allen-Cahn equation. However, proving that the formal procedure described above actually defines a well-posed evolution (e.g. for almost all initial configurations) and investigating its statistical properties after many coarsening iterations is a non-trivial task, which has not been accomplished so far.
Instead, the coarsening model has been studied in the mean field approximation, which consists in merging the minimal interval not with its true neighbors, but with two intervals chosen at random in the configuration {Ij}j∈Z. This approximation is valid provided the lengths of consecutive intervals stay uncorrelated during the coarsening process, see [BDG94] for an argument indicating that the correlations indeed disappear if the number of intervals tends to infinity.
Under this assumption, it is possible to write a closed evolution equation for the distribution f(t, x) (per unit length) of intervals of length x ≥ 1 at time t [CaP92].
Denoting by N(t) = R∞
0 f(t, x) dx the total number of intervals per unit length, and by L(t) the length of the smallest interval, the equation reads
∂tf(t, x) = L(t)f˙ (t,L(t)) N(t)2
Z x−L(t) 0
f(t, y)f(t, x−y−L(t)) dy−2f(t, x)N(t)
!
, (1.1) for x ≥ L(t), whereas f(t, x) = 0 for x < L(t) by the definition of L(t). By construc- tion, N(t) decreases with time, while the total length of the intervals R∞
0 xf(t, x) dx is
conserved.
We prefer to work with the distribution density ρ(t, x) =f(t, x)/N(t), which satisfies ρ(t, x) = 0 for x < L(t) and the normalization R∞
0 ρ(t, x) dx= 1 for all t. The evolution equation forρ reads
∂tρ(t, x) = ˙L(t)ρ(t,L(t))
Z x−L(t) 0
ρ(t, y)ρ(t, x−y−L(t)) dy for x≥ L(t). (1.2) Of course, systems (1.1) and (1.2) are equivalent. In particular, once the densityρ(t, x) is known, the total number N(t) can be recovered by solving the ordinary differential equa- tion ˙N(t) =−2 ˙L(t)ρ(t,L(t))N(t), and the distributionf(t, x) is then given byN(t)ρ(t, x).
It is important to note that equations (1.1), (1.2) are invariant under reparametriza- tions of time. As a consequence, the minimal length L(t) is not determined by the initial data, but can be prescribed to be an arbitrary (increasing) function of time. In [CaP92], the authors define an “intrinsic time” by imposing the relation f(t,L(t)) ˙L(t) = 1, which means that the number of merging events per unit time is constant. We find it more convenient to use the “coarsening time” defined by the simple relation L(t) = t. In other words, we choose to parameterize the coarsening process by the length of the smallest remaining interval, forgetting about how much physical time elapses between or during the merging events. With our choice, equation (1.2) becomes
∂tρ(t, x) =ρ(t, t) Z x−t
0
ρ(t, y)ρ(t, x−y−t) dy for x≥t. (1.3) Since we do not allow for intervals of length smaller than 1, we impose our initial condition at timet = 1: ρ(1, x) =ρ1(x).
The aim of this paper is to show that the dynamics of (1.3) can be completely under- stood using a global linearization transformation. As a consequence, we are able to prove that solutions of (1.3) satisfying R∞
0 xρ(t, x) dx < ∞ approach a non-trivial self-similar profile as t→ ∞. To achieve this goal, we first rewrite (1.3) in similarity coordinates by setting
ρ(t, x) = 1
tη(logt, x/t), or η(τ, y) = eτρ(eτ,eτy),
where τ = logt ≥ 0 and y = x/t ∈ [1,∞). Then the rescaled density η(τ,·) lies in the time-independent space
P=n
η∈L1((1,∞),R+)
Z ∞ 1
η(y) dy= 1o
, (1.4)
which is a closed convex subset of L1((1,∞)). Moreover, (1.3) is transformed into the autonomous evolution equation
∂τη(τ, y) =∂y y η(τ, y)
+η(τ,1) Z y−2
1
η(τ, z)η(τ, y−z−1) dz for y≥1. (1.5) In Section 3 we show that, for all initial data η0 ∈ P, (1.5) has a unique global solution η∈C0([0,∞),P) with η(0) =η0.
We now define a nonlinear map N :P→L1loc([1,∞),R+) by N =F−1◦φ◦ F,
where F is the Fourier transform and φ(z) = 12 log1+z1−z. If η(τ,·) is a solution of (1.5) in P, a direct calculation reveals that w(τ,·) = N(η(τ,·)) satisfies the linear equation
∂τw(τ, y) =∂y(yw(τ, y)). As a consequence, w(τ, y) = (Sτw0)(y) =
eτw0(eτy) if y≥1, 0 if y <1,
where w0 = N(η0). It follows that any solution η ∈ C0([0,∞),P) of (1.5) satisfies N(η(τ)) =SτN(η0) for allτ ≥0. In other words, the nonlinear evolution defined by (1.5) isconjugated(via the mapN) to the linear semigroup (Sτ). Thus, the difficulty of solving (1.5) is carried over to the study of the mappingN and of its inverseN−1 =F−1◦φ−1◦F. Although the properties of these maps are not fully understood, it possible to obtain some information on them using the analyticity properties of the Fourier-Laplace transform.
In Section 4 we investigate the steady states of (1.5), which form a one-parameter family{ηθ∗}θ∈R. Hereηθ∗ =N−1(θ2w∗), wherew∗(y) =y−11{y≥1}. More explicitly, we have
ηbθ∗(ξ) = (Fηθ∗)(ξ) = tanhθ
2E1(iξ)
for ξ∈R, (1.6)
where E1 is the exponential integral [AS72]. We prove thatη∗θ ∈Pif and only if θ∈(0,1].
Moreover, η1∗(y) decays exponentially as y → ∞, while η∗θ(y)∼ y−(1+θ) if 0 < θ < 1. In particular, η1∗ is the only steady state for which the average lengthR∞
1 yη1∗(y) dy is finite.
Finally, Section 5 is devoted to the convergence results. If the initial data η0 ∈ P satisfy yγη0 ∈ L2((1,∞)) for some γ > 3/2 (so that R∞
1 yη0(y) dy < ∞), we prove that the corresponding solution of (1.5) converges exponentially to the steady state η1∗:
kyγ−1(η(τ)−η1∗)kL2((1,∞)) =O(e−(γ−3/2)τ) for τ → ∞.
In terms of the original variables, this shows that the density ρ(t, x) asymptotically approaches the self-similar solution t−1η1∗(x/t) of (1.3). Moreover, the remainder is O(t−(γ−3/2)), so that the convergence is very fast if γ is large, i.e., the initial data decay rapidly at infinity. Similarly, if 0< θ <1 and if η0 ∈P satisfies yγ(η0−νη∗θ)∈L2((1,∞)) for someγ > θ+ 1/2 and someν > 0, we prove that the solution of (1.5) with initial data η0 converges to the steady stateηθ∗.
To conclude this section, we briefly comment on previous results and possible general- izations. The mean field equations (1.1) and especially the self-similar solutions (1.6) can be found in the physics literature [NaK86, DGY91, BDG94, RuB94, BrD95]. The first mathematical work is [CaP92], where the authors prove the existence of global solutions to (1.1). They also show that the profile η∗1 is a positive function (a crucial property that is often tacitly assumed!) and study its asymptotic behavior as y → ∞. Our main contribution is the introduction of the linearization transformation N which allows to prove the convergence results. We also extend the analysis of [CaP92] to the equilibria ηθ∗ with 0< θ <1.
The “two-sided” coarsening model discussed in this introduction is clearly not the most general system to which our analysis applies. For instance, we can consider the
“one-sided” variant in which the minimal interval is merged with one of its neighbors only [CaP00]. More generally, we can assume that, forj = 1, . . . , N, the minimal interval has a probability pj of being merged withj of its neighbors, where p1+· · ·+pN = 1. In the mean field approximation, this leads to an evolution equation similar to (1.3), where the quadratic convolution in the right-hand side is replaced by a more general convolution polynomial. Except for a modified definition of the mapping N, this extension does not affect our analysis in any essential way. Therefore, in the rest of this paper, all results will be stated and proved in this general situation.
Acknowledgments. The authors are grateful for financial support through the French- German grant: PROCOPE 00307 TK, Attractors for Extended Systems.
2 The coarsening equation and its solution
As is explained in the introduction, we shall study a general coarsening model for which the number of intervals involved in each merging event is not necessarily fixed. Instead, we allow for some randomness by choosing nonnegative real numbersp1, . . . , pN satisfying p1 +· · ·+pN = 1, where pj is interpreted as the probability for an interval of minimal length to merge with j other intervals. We define the polynomial
Q(z) = XN
j=1
pjzj,
which satisfies Q(1) = 1. The original coarsening model related to the Allen-Cahn equa- tion corresponds to the particular case where Q(z) =z2.
If ρ∈L1(R), we set
Q[ρ] = XN
j=1
pjρ∗j, (2.1)
where ρ∗j = ρ∗ρ∗ · · · ∗ρ (j factors) and ∗ denotes the convolution product in L1(R).
In particular, we have R∞
0 Q[ρ](x) dx =Q(R∞
0 ρ(x) dx). In what follows, we shall mainly use the space P of probability densities defined by (1.4). Any ρ ∈P can be extended to the whole real line by setting ρ(x) = 0 for x < 1. This natural extension, still denoted by ρ, will be used in the sequel without further mention. As an example of this abuse of notation, if ρ∈P, we have Q[ρ]∈P and supp(Q[ρ])⊂[2,∞) (here and in the sequel, we denote by supp(f) the support of a function f).
The problem we are interested in can now be stated as follows. Given ρ1 ∈ P, find a density ρ : [1,∞)2 → R+ satisfying ρ(1, x) = ρ1(x) for x ≥ 1, ρ(t, x) = 0 for 1 ≤ x < t, and
∂tρ(t, x) =ρ(t, t)Q[ρ(t,·)](x−t) forx≥t ≥1. (2.2) If Q(z) =z2, the evolution equation (2.2) reduces to (1.3).
By assumption, the densityρ(t, x) is nonzero only in the sector{(t, x)∈R2|1≤t ≤x}, where it satisfies (2.2). An important role will be played by the values of ρ on the boundaries of this domain, namely the initial densityρ1 and the trace ofρon the diagonal x=t, which we denote by α:
α(t) = ρ(t, t) for t≥1.
Any sufficiently smooth solution of (2.2) satisfies ρ(t,·) ∈ P for all t ≥ 1 provided ρ1 ∈ P. Indeed, it is obvious from (2.2) that ρ stays nonnegative. Moreover, if m(t) = R∞
t ρ(t, x) dx, a direct calculation shows that d
dtm(t) =α(t) Q(m(t))−1
fort ≥1. (2.3)
Therefore, ifm(1) = 1, thenm(t) = 1 for allt≥1.
A very remarkable property of equation (2.2) is that it can be explicitly solved using Fourier (or Laplace) transform. Ifρ ∈P, we define
ρ(ξ) = (Fb ρ)(ξ) = Z ∞
1
e−iξxρ(x) dx for ξ ∈R.
Then ρb∈ C0(R,C) satisfies ρ(0) = 1,b |ρ(ξ)|b <1 for all ξ 6= 0, andρ(ξ)b →0 as ξ→ ±∞.
Moreover, ρbis a positive definite function (in the sense of Bochner). Since supp(ρ) ⊂ [1,∞), the Fourier transform ρbcan be continuously extended to the lower complex half plane
L− ={ξ∈C| Imξ≤0}.
This extension (still denoted byρ) is analytic in the interior ofb L− and satisfies the bound
|ρ(ξ)| ≤b eImξ for all ξ ∈L−.
Remark. The closely related Laplace transform is defined by ρ(p) = (Lρ)(p) =e
Z ∞ 1
e−pxρ(x) dx for Rep≥0,
so thatρ(p) =e ρ(−ip). In the sequel, we prefer using Fourier transform instead of Laplaceb because the inversion formula is more natural.
Applying Fourier transform to (2.2) and using the fact that convolutions are turned into multiplications, we find the equation
∂tρ(t, ξ) =b α(t) e−iξt
Q(ρ(t, ξ))b −1
for t ≥1, (2.4)
where α(t) = ρ(t, t). To solve (2.4), we introduce the nonlinear complex transformation φ defined by
φ0(z) = 1
1−Q(z) , φ(0) = 0. (2.5)
Remark that φ0(z) = P∞
k=0[Q(z)]k, so that φ has a power series expansion with non- negative coefficients whose radius of convergence is equal to 1. In particular, the map
φ : [0,1) → [0,∞) is one-to-one and onto. Let ψ = φ−1 be the inverse map, which satisfies
ψ0(w) = 1−Q(ψ(w)), ψ(0) = 0. (2.6)
By construction,ψ is analytic in a neighborhood of the real positive axis. In the particular case where Q(z) =z2, one finds
φ(z) = 1
2 log 1 +z
1−z and ψ(w) = tanh(w).
Applying the nonlinear transformationφ simplifies equation (2.4) a lot. The function w(t, ξ) =b φ(ρ(t, ξ)), which is defined at least for Imb ξ < 0, satisfies the differential equation
∂tw(t, ξ) =b −α(t)e−iξt for t ≥1, which has the explicit solution
w(t, ξ) =b w(1, ξ)b − Z t
1
α(s)e−iξsds for t≥1 and Imξ <0. (2.7) Remark that |ρ(t, ξ)| ≤b etImξ for all t ≥ 1 and all ξ ∈ L−, because ρ(t,·) ∈ P and supp(ρ(t,·)) ⊂ [t,∞). Since φ(z) = z +O(|z|2) as z → 0, it follows that |w(t, ξ)|b =
|φ(ρ(t, ξ))| →b 0 as t → ∞ if Imξ < 0. Thus, taking the limit t → ∞ in (2.7), we find w(1, ξ) =b R∞
1 α(t) e−iξtdt, which in turn implies b
w(t, ξ) = Z ∞
t
α(s)e−iξsds for t≥1 and Imξ <0. (2.8) This formula has a very nice interpretation. Let N be the nonlinear transformation defined (at least formally) by
N =F−1◦φ◦ F or N−1 =F−1◦ψ◦ F. (2.9) Settingt = 1 in (2.8), we obtain φ(ρb1) = α, that isb α =N(ρ1). In other words, the trace α(t) = ρ(t, t) is obtained from the initial densityρ1(x) =ρ(1, x) by applying the nonlinear map N. Moreover, ifU(t) is the linear operator defined for t≥1 by
(U(t)w)(s) =1{s≥t}w(s) =
0 if s < t,
w(s) if s≥t, (2.10)
then (2.8) reads w(t,b ·) =φ(ρ(t,b ·)) =F(U(t)α), which means N(ρ(t,·)) =U(t)α. There- fore, the solution of (2.2) satisfies
N(ρ(t,·)) =U(t)N(ρ1) for t≥1. (2.11) This shows that the dynamics of the nonlinear system (2.2) is conjugated via the nonlinear mapping N to the linear evolution U. Since N(ρ1) is the trace function defined by α(t) = ρ(t, t), it is very natural that the evolution of α is obtained just by cutting off the history in [1, t).
It is not difficult to show that the map N is well-defined on the space P, cf. (1.4):
Proposition 2.1 If ρ∈P, then N(ρ)∈L1loc([1,∞),R+), and the mapping ρ7→ N(ρ) is one-to-one.
Proof. For ρ ∈ P we construct w = N(ρ) as follows. Define wb : L−∗ → C by w(ξ) =b φ(ρ(ξ)), whereb L−∗ = L− \ {0} . We recall that ρbis continuous on L−, analytic in the interior of L−, and that |ρ(ξ)|b < 1 for ξ 6= 0. Since φ is analytic in the unit disk of C, it follows that wb is continuous on L−∗ and analytic in the interior of L−. Moreover,
|w(ξ)| ≤b φ(|ρ(ξ)|)b ≤ φ(eImξ), hence |w(ξ)|b = O(eImξ) as Imξ → −∞. These properties imply (see [Sch66], Ch. VIII) that wb is the Fourier transform of a uniquely determined distribution w∈ D0(R) with support in [1,∞).
The injectivity of N follows from the facts that the mapping φ:{z | |z|<1} →C is locally injective (as φ0(z) = 1/(1−Q(z))6= 0) and thatφ : [0,1)→R is globally injective (as φ0(s)≥ 1 for s ∈ [0,1)). If N(ρ1) = N(ρ2), then, by the above, we have φ(ρb1(ξ)) = φ(ρb2(ξ)) for ξ ∈ L−∗. This proves ρb1(−ip) = ρb2(−ip) for p > 0, as ρbj(−ip) ∈ [0,1). By continuity of ρbj and local invertibility we obtain ρb1 =ρb2 on L−∗, and hence ρ1 =ρ2.
To prove w = N(ρ) ∈ L1loc([1,∞)), choose any ε > 0 and consider the distribution wε : x 7→ e−εxw(x). It belongs to S0(R) (the space of tempered distributions) and its Fourier transform satisfies
wbε(ξ) = w(ξ−iε) =b φ(ρ(ξ−iε)) for Imb ξ ≤0.
Now we observe that ρ(ξ−iε) =b ρbε(ξ), where ρε(x) = e−εxρ(x). Since kρεkL1 ≤ e−ε < 1, the seriesP∞
k=1 φ(k)(0)
k! ρ∗kε converges inL1(R) to some functionWε∈L1([1,∞),R+). (Here we use the crucial fact that φ(k)(0)≥0 for all k ∈N.) By construction,
Wcε(ξ) = X∞ k=1
φ(k)(0)
k! ρbε(ξ)k
=φ(ρ(ξ−iε)) =b wbε(ξ) for Imξ ≤0,
giving wε=Wε∈L1((1,∞),R+), and hence w:x7→eεxwε(x) lies in L1loc([1,∞),R+).
Remarks.
1. Under the assumptions of Proposition 2.1, one has that w = N(ρ) ∈ S0(R), i.e., w is a tempered distribution. In fact, there exists a constant C > 0 such that |w(ξ)|b =
|φ(ρ(ξ))| ≤b Cmax{1,−log|ξ|} for ξ 6= 0, see the proof of Proposition 5.1 below. This means that the singularity of w(ξ) atb ξ = 0 is (not worse than) logarithmic.
2. More information on N can be extracted from the proof of Proposition 2.1. For instance, if ρ∈P, then N(ρ)(x) =ρ(x) for almost all x∈(1, n+1), where
n= min
j ∈ {1, . . . , N}pj >0 ≥1 (2.12) is the largest integer such that |Q(z)| =O(|z|n) as z →0. Indeed, in view of (2.5), one has φ(z) =z+O(|z|n+1) as z →0. It follows that
Wε =ρε+ X∞ k=n+1
φ(k)(0) k! ρ∗kε ,
where the second term in the right-hand side is supported in the interval [n+1,∞). Thus Wε = ρε almost everywhere in [1, n+1], which proves the claim. Similarly, using the
observation that supp(ρ∗kε ) ⊂ [k,∞), it is easy to show that, if ρ : [1,∞) → R+ is continuous, so is N(ρ).
The formula (2.11) is very nice, but does not provide an effective method for solving the Cauchy problem associated with (2.2). Indeed, Proposition 2.1 does not give a sufficient characterization of the set N(P), which is also the domain of N−1. It is not even clear a priori that this set in left invariant by the linear evolution U(t). For this reason, we shall use standard PDE techniques to prove existence of solutions to (2.2) in the next section.
But the representation (2.11) will be very useful to find self-similar solutions of (2.2) in Section 4, and to study their stability in Section 5.
3 The Cauchy problem for the rescaled system
The evolution equation (2.2) is not autonomous, and it is defined on the time-dependent domain{x∈R+|x≥t}. These drawbacks are eliminated if we rescale the densityρ(t, x) by setting
ρ(t, x) = 1
tη(logt, x/t) for x≥t≥1, (3.1) or equivalently
η(τ, y) = eτρ(eτ,eτy) for τ ≥0, y ≥1. (3.2) In what follows, we denote by τ = logt andy =x/t the new time and space coordinates.
The rescaled density η(τ,·) now belongs to the fixed space P defined in (1.4). Moreover, it satisfies the autonomous evolution equation
∂τη(τ, y) =∂y(y η(τ, y)) +β(τ)Q[η(τ,·)](y−1) for y≥1, (3.3) whereβ(τ) = η(τ,1) is the new trace which relates toα(t) viaβ(τ) = eτα(eτ). The initial condition for (3.3) isη(0, y) =η0(y), where η0 =ρ1 ∈P.
The nonlinearity in (3.3) has the form β(τ)T1Q[η(τ)], where T1 : P → P is the shift operator defined by
(T1η)(y) =
η(y−1) if y≥2;
0 if y <2. (3.4)
In particular, for all η ∈ P, the support of T1Q[η] is contained in [2,∞), or even in [n+1,∞), where n ≥ 1 is defined in (2.12). Thus, any solution of (3.3) satisfies the linear equation ∂τη = ∂y(yη) in the strip {(τ, y)|τ ≥ 0,1 ≤ y ≤ 2}. It follows that η(τ, y) = eτ−τ0η(τ0,eτ−τ0y) for all τ ≥τ0 ≥0 and ally ≥1 such that eτ−τ0y ≤2. Setting y= 1, we obtain the important relation
β(τ) = eτ−τ0η(τ0,eτ−τ0) for 0≤ τ−τ0 ≤log 2, (3.5) which means that the traceβ(τ) forτ ∈[τ0, τ0+ log 2] can be determined from the solution η(τ0,·). This formula will be useful to define the traceβproperly when the solutionη(τ,·) of (3.3) is not continuous. For instance, if η(τ,·)∈P for all τ ≥0 and if β satisfies (3.5), then β ∈L1loc([0,∞),R+).
The main purpose of this section is to show that (3.3) defines a well-posed evolution in the space P. To do this, we consider the associated integral equation
η(τ) =Sτη0+ Z τ
0
β(s)Sτ−sT1Q[η(s)] ds for τ ≥ 0, (3.6) where (Sτ)τ≥0 is the linear semigroup on P defined by
(Sτη)(y) =
eτη(eτy) if y≥1;
0 if y <1. (3.7)
To formulate our convergence results in Section 5, we shall need some weighted Lp spaces which we now introduce. For p∈[1,∞) and γ ≥0, we denote by Lpγ the function space
Lpγ ={w∈L1loc([1,∞),R)| kwkp,γ <∞}, (3.8) where
kwkp,γ =kyγwkLp = Z ∞
1
(yγ|w(y)|)pdy 1/p
.
When γ = 0, we simply write Lp instead ofLp0 and kwkp instead of kwkp,0. Remark that Lpγ ,→L1 if and only ifγ >1−1/p(whenp >1) orγ ≥0 (whenp= 1). In what follows, we shall often restrict ourselves to such values of p, γ.
We first give a few basic estimates on the semigroup (Sτ) and the nonlinearityQacting onLpγ.
Lemma 3.1 Let p ∈ [1,∞) and γ ≥ 0. Then (3.7) defines a strongly continuous semi- group (Sτ)τ≥0 in Lpγ, and
kSτηkp,γ ≤e−τ(γ−1+1/p)kηkp,γ, (3.9) for all η ∈Lpγ and all τ ≥0. Moreover, equality holds in (3.9) if and only if η(y) = 0 for almost all y∈[1,eτ].
Lemma 3.2 Let Q be the nonlinear map defined by (2.1).
a) If η ∈L1, then Q[η]∈L1 and kQ[η]k1 ≤Q(kηk1). If η,η˜∈L1, then kQ[η]−Q[˜η]k1 ≤Q0(r)kη−ηk˜ 1,
where r= max{kηk1,kηk˜ 1}. Finally, if η∈P, then Q[η]∈P.
b) Let p ∈ [1,∞) and γ > 1−1/p. If η ∈ Lpγ, then Q[η] ∈ Lpγ, and there exists C > 0 (independent of η) such that
kT1Q[η]kp,γ ≤CQ0(kηk1)kηkp,γ. (3.10) If η,η˜∈Lpγ and R = max{kηkp,γ,k˜ηkp,γ}, then
kT1Q[η]−T1Q[˜η]kp,γ ≤CQ0(R)kη−ηk˜ p,γ.
Proof. Estimate (3.9) is a straightforward calculation, and the proof of Lemma 3.2 will be outlined in Appendix C.
We are now ready to state the main result of this section:
Theorem 3.3 For any η0 ∈ L1((1,∞),R) with kη0k1 ≤ 1, equations (3.6), (3.5) have a unique global solution η ∈ C0([0,∞), L1), which satisfies kη(τ)k1 ≤ 1 for all τ ≥ 0. In addition,
1) if η0 ∈P, then η(τ)∈P for all τ ≥0;
2) if η0 ∈Lpγ for some p≥1 and some γ >1−1/p, then η ∈C0([0,∞), Lpγ).
Proof. Fix η0 ∈B1, where B1 ={η∈L1| kηk1 ≤1}. Setting τ0 = 0 in (3.5), we obtain β(τ) = eτη0(eτ) for 0≤τ ≤log 2. (3.11) The first step is to show that (3.6),(3.11) have a unique solution η ∈C0([0,log 2], L1).
Let q =Q0(1)≥1, and let T = (log 2)/m, where m ∈N∗ is sufficiently large so that, for all k = 1, . . . , m,
Z kT (k−1)T
es|η0(es)|ds < 1
q. (3.12)
We introduce the Banach space X =C0([0, T], L1) equipped with the norm kηkX = sup
0≤τ≤T
kη(τ)k1.
Let B ={η ∈X| kηkX ≤1}, and let F :X 7→X be the nonlinear map defined by (F[η])(τ) =Sτη0 +
Z τ 0
β(s)Sτ−sT1Q[η(s)] ds for 0≤τ ≤T,
whereβ(s) is given by (3.11). We claim that F(B)⊂B and thatF is a strict contraction in B. Indeed:
a) Assume that η∈B. Using Lemmas 3.1 and 3.2, we find, for all τ ∈[0, T], k(F[η])(τ)k1 ≤ kSτη0k1 +
Z τ 0
|β(s)|kSτ−sT1Q[η(s)]k1ds
= Z ∞
1
eτ|η0(eτy)|dy+ Z τ
0
|β(s)|kT1Q[η(s)]k1ds
= Z ∞
eτ
|η0(y)|dy+ Z τ
0
es|η0(es)|kQ[η(s)]k1ds (3.13)
≤ Z ∞
eτ
|η0(y)|dy+Q(kηkX) Z eτ
1
|η0(y)|dy ≤ 1, since Q(kηkX)≤Q(1) = 1 and kη0k1 ≤1. This shows that F(B)⊂B.
b) Ifη,η˜∈B, then for all τ ∈[0, T], k(F[η])(τ)−(F[˜η])(τ)k1 ≤
Z τ 0
|β(s)|kSτ−s(T1Q[η(s)]−T1Q[˜η(s)])k1ds
= Z τ
0
es|η0(es)|kQ[η(s)]−Q[˜η(s)])k1ds
≤ Z τ
0
es|η0(es)|Q0(1)kη(s)−˜η(s)k1ds
≤ qZ T
0
es|η0(es)|ds
kη−˜ηkX. In view of (3.12), this shows that F is a strict contraction in B.
Let η ∈ X be the unique fixed point of F in the ball B. Then η satisfies (3.6), and using Gronwall’s lemma it is readily verified thatη is in fact the unique solution of (3.6) in the whole space X =C0([0, T], L1). Repeating the same argument m times (where m is such that (3.12) holds), we conclude that equations (3.6), (3.11) have a unique solution η∈C0([0,log 2], L1), which satisfieskη(τ)k1 ≤1 for all τ ∈[0,log 2]. Moreover, it is clear that (3.5) holds for all τ0 ∈[0,log 2] and almost all τ ∈[τ0,log 2].
For τ ∈ [0,log 2], let Ξτ : B1 → B1 be the nonlinear map defined by Ξτη0 = η(τ), whereη(τ) is the solution of (3.6) we have just constructed. Then it is easy to verify that Ξτ1+τ2 = Ξτ1◦Ξτ2 for 0≤τ1+τ2 ≤log 2. It follows that the family (Ξτ) can be extended to a continuous semiflow (Ξτ)τ≥0. By construction, if η0 ∈B1 and if we set η(τ) = Ξτη0
for all τ ≥0, thenη ∈C0([0,∞), L1) is the unique solution of (3.6), (3.5), andη(τ)∈B1
for all τ ≥0. This proves the first part of Theorem 3.3.
Assume now that η0 ∈P. Keeping the same notations as above, we define B˜ ={η∈X|η(τ)∈Pfor all τ ∈[0, T]}.
In particular, ˜B is a closed subset of B, as P is closed in B1 ⊂ L1. If η ∈ B, it is clear˜ that (F[η])(τ) ∈ L1((1,∞),R+) for all τ ∈ [0, T], and that all inequalities in (3.13) can be replaced by equalities. Thus F( ˜B)⊂B, hence the solution˜ η∈C0([0,∞), L1) of (3.6) satisfies η(τ)∈P for all τ ∈[0, T]. Proceeding as above, we then show that η(τ)∈P for allτ ∈[0,log 2], hence for all τ ≥0. This proves assertion 1) in Theorem 3.3.
Finally, assume thatη0 ∈Lpγ for somep≥1 and someγ >1−1/p, and thatkη0k1 ≤1.
Using Lemmas 3.1, 3.2 and a fixed point argument as before, it is straightforward to show that the solution η ∈ C0([0,∞), L1) of (3.6) satisfies η ∈ C0([0, T], Lpγ) for some T > 0 (depending on η0). Let
T∗ = sup
T >0
η∈C0([0, T], Lpγ) ∈(0,∞].
We claim that T∗ = ∞. Indeed, assume on the contrary that 0 < T∗ < ∞. Since kη(τ)k1 ≤1 for allτ ≥0, it follows from (3.6), (3.9), (3.10) that
kη(τ)kp,γ≤ kη0kp,γ+Cq Z τ
0
|β(s)|kη(s)kp,γds for 0≤ τ < T∗.
Using Gronwall’s lemma and the fact thatβ∈L1loc([0,∞)), we deduce thatkη(τ)kp,γ ≤C0 for all τ ∈ [0, T∗). In view of (3.6), (3.10), this in turn implies that η(τ) has a limit in
Lpγ asτ %T∗, giving η∈C0([0, T∗], Lpγ). Since we have a local existence result in Lpγ, we conclude that η ∈C0([0, T], Lpγ) for some T > T∗, which contradicts the definition of T∗. This proves assertion 2) in Theorem 3.3.
The nonlinear mapN introduced in the previous section can also be used to linearize (3.3). Indeed, the Fourier transforms of ρ and η are related via ρ(t, ξ) =b bη(logt, tξ), so that (3.1) is just a rescaling of the Fourier variable ξ. As is clear from (2.9), this transformation commutes with the action of N. Thus, if ρ is a solution of (2.2) with initial dataρ1 and if ηis the corresponding solution of (3.3) given by (3.2), it follows from (2.11) that
1
tN η(logt,·)
(x/t) =N(η0)(x) for x≥t≥1, (3.14) where η0 =ρ1. Setting τ = logt and y=x/t, we obtain the representation formula
N(η(τ)) =SτN(η0) for τ ≥0, (3.15) where (Sτ) is the linear semigroup (3.7). The last result of this section shows that this formula is indeed correct:
Proposition 3.4 Let η0 ∈P, and let η ∈ C0([0,∞),P) be the solution of (3.6) given by Theorem 3.3. Then N(η(τ)) = SτN(η0) for all τ ≥0.
Proof. We establish the formula by returning to the unscaled variables (t, x) and by showing that the formal steps of Section 2 can be made rigorous for the solutions of (3.3). Define ρ : [1,∞)2 → R+ by ρ(t, x) = 1tη(logt, x/t) if x ≥ t ≥ 1 and ρ(t, x) = 0 if 1≤x < t. Thenρ∈C0([1,∞),P), and rescaling (3.6) we find
ρ(t) =U(t)
ρ1+ Z t
1
α(s)TsQ[ρ(s)] ds
for t ≥1, (3.16)
where ρ1 =η0 ∈P,α(t) = 1tβ(logt),U(t) is the linear operator (2.10), and Ts is the shift operator defined as in (3.4). To simplify the notation, we set f(s, x) = (TsQ[ρ(s)])(x).
Thenf ∈C0([1,∞), L1), so that (s, x)7→α(s)f(s, x)∈L1loc([1,∞), L1). By construction, the trace α satisfies the identity
α(t) =ρ1(t) + Z t
1
α(s)f(s, t) ds for a.a. t ≥1.
We now apply the Fourier transform to (3.16). For any ξ∈L− and anyt ≥1, we find b
ρ(t, ξ) = Z ∞
t
ρ1(x)e−iξxdx+ Z ∞
t
n Z t
1
α(s)f(s, x) dso
e−iξxdx.
Since ρ1 ∈ P, the first term in the right-hand side is absolutely continuous with respect tot, and
∂t
Z ∞ t
ρ1(x)e−iξxdx=−ρ1(t)e−iξt for a.a. t ≥1.
The second term can be decomposed ash1(t, ξ)−h2(t, ξ), where h1(t, ξ) =
Z ∞ 1
n Z t
1
α(s)f(s, x) dso
e−iξxdx = Z t
1
α(s)fb(s, ξ) ds, h2(t, ξ) =
Z t 1
n Z t
1
α(s)f(s, x) dso
e−iξxdx.
Clearly, h1(t, ξ) is absolutely continuous with respect to t, and
∂th1(t, ξ) =α(t)fb(t, ξ) =α(t)e−iξtQ(ρ(t, ξ)) for a.a.b t ≥1.
Next, since f(s, x) = 0 for x < s, we have Rt
1α(s)f(s, x) ds = Rx
1 α(s)f(s, x) ds, and this expression is a locally integrable function of x. It follows that h2(t, ξ) is absolutely continuous with respect to t, and
∂th2(t, ξ) = e−iξt Z t
1
α(s)f(s, t) ds for a.a. t≥1.
Summarizing, we have shown that, for any ξ∈ L−, the Fourier transform ρ(t, ξ) is abso-b lutely continuous with respect to t and satisfies
∂tρ(t, ξ) =b −e−iξt
ρ1(t) + Z t
1
α(s)f(s, t) ds
+α(t)e−iξtQ(ρ(t, ξ))b
= α(t) e−iξt
Q(ρ(t, ξ))b −1
for a.a. t≥1.
This gives (2.4). Now, proceeding exactly as in Section 2, we deduce that (2.8) holds for all t ≥ 1 if Imξ < 0, and this in turn is equivalent to (2.11). Finally, using the transformation (3.14) we obtain (3.15).
4 Properties of the steady states
This section is devoted to the time-independent solutions of (3.3) in the space P defined by (1.4).
Definition. We say thatη0 ∈Pis asteady state of (3.3) if the solutionη ∈C0([0,∞),P) of (3.6) given by Theorem 3.3 satisfies η(τ) =η0 for all τ ≥0.
The steady states of (3.3) will also be called “equilibria” or “stationary solutions”.
Lemma 4.1 If η0 ∈P is a steady state of(3.3), there exists β ≥0 such that η0(y) =β/y for almost all y∈[1,2].
Proof. If η(τ) ≡ η0, (3.6) implies that η0(y) = eτη0(eτy) for all τ ∈ [0,log 2] and a.a. y∈[1,2 e−τ], because the nonlinearity in (3.6) vanishes identically for such values of τ, y. We define F :x7→Rex
1 η0(y) dy≥0 and obtain
F(x+y) =F(x) +F(y) for x, y ≥0 and x+y≤log 2.
Since F is continuous, we conclude that F(x) = βx for some β ≥ 0. Differentiating implies β= exη0(ex) for a.a. x∈[0,log 2] which gives the desired result.
Let η0 ∈ P be a steady state. Since η0 coincides almost everywhere in [1,2] with a continuous function, the constant β in Lemma 4.1 can be identified with η0(1). Clearly, the trace function defined by (3.5) satisfies β(τ) = β for all τ ≥ 0. In particular, the integral equation (3.6) reduces to
η0 =Sτη0+β Z τ
0
SsT1Q[η0] ds for τ ≥0. (4.1) From η0 ∈P we now conclude thatβ >0.
On the other hand, if η0 ∈P and w=N(η0), it follows from Propositions 2.1 and 3.4 that η0 is a steady state if and only if Sτw=w for all τ ≥0. In view of (3.7), this is the case if and only if there exists β0 ∈R such that w=β0w∗, where
w∗(y) =
1/y if y≥1,
0 if y <1. (4.2)
But since w(y) =η0(y) for a.a. y∈[1,2] (see Remark 2 after Proposition 2.1), we neces- sarily have β0 =β =η0(1).
Finally, since equilibria are time-independent solutions of (3.3), we certainly expect them to solve the ordinary differential equation
(yη)0(y) +β(T1Q[η])(y) = 0 for y≥1, η(1) =β. (4.3) Remark that the initial valueβalso appears as a parameter in front of the nonlinear term.
It is not difficult to show that (4.3) has global solutions:
Lemma 4.2 For any β ∈R, equation (4.3) has a unique global solution η: [1,∞)→R. Proof. For any k ∈ N, letIk = [kn+ 1,(k+1)n+ 1], where n ∈ N∗ is defined in (2.12).
For any η∈L1loc([1,∞),R), the nonlinear term (T1Q[η])(y) only depends on the values of η(z) for z ≤ y−n. In particular, (T1Q[η])(y) = 0 for y ≤ n+ 1, so that any solution of (4.3) satisfies η(y) =β/y fory∈I0 = [1, n+ 1]. Using this information, one can compute (T1Q[η])(y) explicitly for y ∈ I1 = [n + 1,2n+ 1], and then solve (4.3) on this interval to determine η(y) for y ∈ I1. By construction, η is smooth on both I0 and I1, but η has a discontinuity of order n at y = n+ 1, in the sense that the derivatives η(k)(y) are continuous fork = 0, . . . , n−1, whereasη(n)(y) has different limits to the left and to the right at y = n+ 1 (if β 6= 0). Iterating this procedure, we find that (4.3) has a unique global solution η∈Cn−1,1([1,∞),R), which satisfies η∈C∞(Ik) for all k ∈N.
The following result shows that equilibria of (3.3) indeed correspond to solutions of the differential equation (4.3).
Proposition 4.3 If η0 ∈P and β >0, the following assertions are equivalent:
a) η0 is a steady state of (3.3) with η0(1) =β.
b) η0 coincides almost everywhere with the solution of (4.3).
c) N(η0) =βw∗.
Proof. We already proved that a) ⇔ c). If η0 ∈ P is a steady state with η0(1) = β, it follows from (4.1) that
Sτη0−η0
τ + β
τ Z τ
0
SsT1Q[η0] ds = 0,
for allτ >0. Using (3.7), it is not difficult to verify that the first term converges to (yη0)0 inD0((1,∞)) asτ →0, while the second one tends to βT1Q[η0] inL1((1,∞)). This shows that (after modification on a set of measure zero) η0 is absolutely continuous on (1,∞) and satisfies the differential equation (4.3) for almost all y > 1. It follows easily that η0
is the solution of (4.3) in the sense of Lemma 4.2. Thus a)⇒b).
Conversely, assume that η0 ∈ P satisfies (4.3). Applying the semi-group Sτ to (4.3) and integrating over τ, we immediately obtain (4.1), which implies that η0 is a steady state. This proves that b)⇒a).
The main goal of this section is to determine for which values ofβ >0 the solution η of (4.3) belongs to P. Our strategy is to use the characterization c) in Proposition 4.3.
Therefore, we are led to study the image of βw∗ under the map N−1, and this requires very precise information on the complex transformations (2.5) and (2.6). The following quantities, related to the polynomial Q(z), will play an important role in the sequel:
q =Q0(1)≥1 and κ= exp Z 1
0
1
1−z − q 1−Q(z)
dz
≤1. (4.4)
Lemma 4.4 Let
Φ(z) = 1−e−qφ(z) for |z|<1,
where φ is defined in (2.5). Then Φ can be extended analytically to a neighborhood of the real positive axis R+. This extension satisfies Φ(z) ≥ 0 and Φ0(z) > 0 for all z ≥ 0.
Moreover, Φ(0) = 0, Φ0(0) =q, Φ(1) = 1, Φ0(1) =κ, and Φ(z)→R as z → ∞, where R= 1 + exp
Z 2 0
1
1−z − q 1−Q(z)
dz− Z ∞
2
q
1−Q(z)dz
. (4.5)
Note that R=∞ if Q(z) =z and 1< R <∞ otherwise.
Proof. Since the polynomial 1−Q(z) has the unique real positive rootz = 1, which is a simple root because Q0(1) =q 6= 0, it is clear that the function
χ(z) = exp Z z
0
1
1−t − q 1−Q(t)
dt
= e−qφ(z)
1−z for |z|<1,
can be extended to an analytic map in a neighborhood of the real positive axis R+. Moreover, χ(0) = 1, χ(1) =κ, and using z−1 = exp(−Rz
2 dt
1−t) shows that (z−1)χ(z)→ R−1 for z → ∞, where R is defined in (4.5). Since Φ(z) = 1−(1−z)χ(z), we conclude that the function Φ has the desired properties. In particular,
Φ0(z) =qχ(z) 1−z 1−Q(z),
so that Φ0(z)>0 for all z≥0.
It follows from Lemma 4.4 that the map Φ : [0,∞) → [0, R) is one-to-one and onto.
Let Ψ = Φ−1 : [0, R)→[0,∞) be the inverse map. Then Ψ(0) = 0, Ψ0(0) = 1/q, Ψ(1) = 1, Ψ0(1) = 1/κ, and Ψ0(u)>0 for allu∈[0, R). By construction,
Ψ(u) =ψ
−1
q log(1−u)
for 0≤u <1. (4.6)
Lemma 4.5 The function Ψ : [0, R) → [0,∞) is absolutely monotone, i.e. Ψ(k)(u) ≥ 0 for all k ∈N and all u∈[0, R). In particular, Ψ can be extended to an analytic function on the disc |u|< R, and there exist nonnegative coefficients (Ψk)k∈N
∗ such that Ψ(u) =
X∞ k=1
Ψkuk for |u|< R.
Proof. Since Ψ = Φ−1, we already know that Ψ is analytic in a neighborhood of [0, R).
We first show by induction that, for all n∈N∗, there exists a polynomialPn such that Ψ(n)(u) = Pn(Ψ(u))
qn(1−u)n for 0< u <1. (4.7) Indeed, differentiating (4.6) and using (2.6), we obtain
Ψ0(u) = 1−Q(Ψ(u))
q(1−u) for 0< u < 1. (4.8) Thus (4.7) holds for n = 1 with P1(z) = 1−Q(z). On the other hand, differentiating (4.7) and using (4.8), we find, for 0< u < 1,
Ψ(n+1)(u) = Pn+1(Ψ(u))
qn+1(1−u)n+1 with Pn+1(z) =Pn0(z)(1−Q(z)) +nqPn(z). (4.9) Therefore, (4.7) is established.
We next show that, for all n∈ N∗, there exists a polynomial Rn(z) with nonnegative coefficients such that
Pn(z) = (1−Q(z))(1−z)n−1Rn(z). (4.10) Obviously, (4.10) holds for n= 1 with R1(z) = 1. Combining (4.9) and (4.10), we obtain the recursion relation
Rn+1(z) =A1(z)R0n(z) +A2(z)Rn(z) + (n−1)A3(z)Rn(z), where the coefficient functions Aj are given by
A1(z) = 1−Q(z) 1−z =
XN j=1
pj
1−zj 1−z , A2(z) = q−Q0(z)
1−z = XN
j=2
jpj1−zj−1 1−z , A3(z) = q
1−z − 1−Q(z) (1−z)2 =
XN
pj
Xj−1 1−zk 1−z .
Because of pj ≥ 0 all A1, A2, A3 are polynomials (in z) with nonnegative coefficients.
Thus, the same property holds for Rn by induction over n.
Since 0 <Ψ(u)<1 and 0 < Q(Ψ(u))<1 for all u ∈ (0,1), it follows from (4.7) and (4.10) that Ψ(n)(u) ≥ 0 for all n ∈ N and all u ∈ (0,1), hence also for u ∈ [0,1]. By a classical result of Bernstein (see [Fe71], Section VII.2), the power series
X∞ k=1
Ψkuk, where Ψk= 1
k!Ψ(k)(0)≥0, (4.11)
converges absolutely and uniformly for |u| ≤ 1, and defines an analytic continuation of Ψ to the unit disk. Moreover, if R1 ≥ 1 denotes the radius of convergence of the series (4.11), it is well-known (see for instance [Ru87], exercise 16.1) that the analytic function defined by (4.11) has a singularity at u=R1. Since Ψ(u)→ ∞ asu%R, it follows that R =R1. This concludes the proof.
Example. To conclude this study of the mappings Φ and Ψ, we give an explicit example of a nonlinearity Q for which these functions can be calculated explicitly. Let Q(z) = (1−a)z+az2, where a ∈[0,1]. The value a = 1 corresponds to the coarsening equation (1.3), whilea= 0 is a particular case of a model studied in [CaP00]. Thenq= 1+a= 1/κ, R = 1 + 1/a, and
φ(z) = 1
1 +alog1 +az
1−z , ψ(w) = 1−e−qw 1 +ae−qw. The auxiliary functions Φ, Ψ are:
Φ(z) = (1+a)z
1 +az , Ψ(u) = u 1+a−au.
We are now ready to state and prove the main result of this section.
Theorem 4.6 (Steady states of (3.3))
Fix θ > 0and let ηθ∗ : [1,∞)→R be the solution of (4.3) with β =θ/q. Then a) η∗θ ∈P if and only if 0< θ ≤1.
b) If θ∈(0,1], η∗θ ∈P is positive and strictly decreasing, so that yηθ∗(y)→0 as y→ ∞.
c) If 0< θ <1, then
y→∞lim y1+θηθ∗(y) = θeθγE
κΓ(1−θ), (4.12)
where Γ is the Gamma function and γE=−Γ0(1)≈0.577216 is Euler’s constant.
d) If θ= 1, then Z ∞
1
yη1∗(y) dy= eγE
κ . (4.13)
Moreover, if degQ >1, there exists λ >0 such that
y→∞lim
logη1∗(y)
y =−λ. (4.14)
For Q(z) =z we have
y→∞lim
logη1∗(y)
ylogy =−1. (4.15)
Remark. It follows from Theorem 4.6 and Proposition 4.3 that (3.3) has auniquesteady state η∗1 ∈P such that R∞
1 yη1∗(y) dy <∞.
Proof. We first show that η∗θ ∈ P if 0 < θ ≤ 1. According to Proposition 4.3, it is sufficient to prove that there exists an element of P (still denoted by ηθ∗) such that N(ηθ∗) = (θ/q)w∗. Since N−1 = F−1 ◦ ψ ◦ F and ψ(w) = Ψ(1−e−qw) by (4.6), this relation is equivalent to
b
η∗θ = Ψ 1−e−θwb∗
, (4.16)
where bηθ∗ =Fηθ∗ and wb∗ =Fw∗. In view of (4.2), wb∗(ξ) =
Z ∞ 1
e−iξy
y dy = E1(iξ), (4.17)
where E1 is the exponential integral, see [AS72]. It is well-known that
E1(z) =−logz−γE+χ(z) for |argz|< π, (4.18) where χ:C→C is an entire function with χ(0) = 0 and χ0(0) = 1. Thus, wb∗ is analytic in the interior of L−, where L− = {ξ ∈ C| Imξ ≤ 0}. Moreover, Re(wb∗(ξ)) → ∞ as ξ →0 within L−.
In Appendix A, we prove that |1−e−θwb∗(ξ)|<1 for allξ ∈L−\ {0} and allθ∈(0,1], see also Figure A.1. From Lemma 4.5, we also know that Ψ is analytic in the disk of radius R >1 centered at the origin. Therefore, the map bηθ∗ defined by (4.16) is continuous over L− (with ηbθ∗(0) = 1) and analytic in the interior of L−. In addition, since |Ψ(u)| ≤ |u|
whenever |u| ≤1, we have the bound
|bηθ∗(ξ)| ≤ |1−e−θwb∗(ξ)| ≤2θ|wb∗(ξ)| for ξ ∈L−\ {0}.
In particular, |ηb∗θ(ξ)|= O(eImξ) as Imξ → −∞. By the Paley-Wiener Theorem (see for instance [Ru87]), we conclude that η∗θ =F−1bηθ∗ ∈L2((1,∞)).
To prove that ηθ∗ is nonnegative, we argue as in [CaP92]. Consider the Laplace transform eηθ∗ = Lηθ∗, which satisfies ηeθ∗(p) = bηθ∗(−ip). As is well-known (see [Fe71], Section XIII.4), positivity of ηθ∗ is equivalent to complete monotonicity of ηe∗θ, namely (−1)kηe∗θ(k)(p)≥0 for all k ∈N and p >0. Recall that
ηeθ∗(p) = Ψ(1−e−θwe∗(p)) = Ψ(1−e−θE1(p)) for p >0. (4.19) We apply Lemma 4.7 below with
f1 :
(0,1) → R,
u 7→ Ψ(1−u); and g1 :
(0,∞) → (0,1), p 7→ e−θE1(p).
By Lemma 4.5, f1 is completely monotone, thus it remains to show that g01 is completely monotone. Observe that g10 =f2◦g2, where f2 : R→ R is defined by f2(w) =θe−w and g2 : (0,∞)→Rby
g2(p) =θE1(p)−log(−E01(p)) =θE1(p) +p+ logp.
